Simplify Calculator

Interactive calculator

Polynomial simplifier

Simplify a bounded one-variable polynomial by expanding products and powers, applying numeric division, combining like terms, and ordering the result.

Ready · Polynomial modeInteractive tool

WorkspaceCalculator controls

Use the controls below to enter, calculate, and review.

Polynomial expression in xUse numbers, x, +, −, ×, ÷ by a number, whole-number powers, and parentheses.

Simplified polynomial5x + 3

Degree: 1.00
Nonzero terms: 2.00

How it was simplified

Combine 2x and 3x into 5x.
Combine −4 and 7 into 3.

This bounded polynomial tool is not an arbitrary CAS. It supports one real variable x, degree 20 or below, and division only by a nonzero numeric constant.

Polynomial rewrite bench

Expand a polynomial without changing the function it represents

Simplification is controlled rewriting: parentheses are distributed, powers are evaluated within a bounded grammar, and coefficients of identical powers combine while the expression’s value remains unchanged for every input.

Step-by-step guide · Polynomial rewrite bench

A frame-width model rewards careful double distribution

Variable scope: one variable, x
Canonical order: descending powers
Equality goal: equivalent expression, not a solved x

Expand the outer product
(x+6)(x+4)=x²+4x+6x+24=x²+10x+24. Every term in one factor multiplies every term in the other.
Expand the opening and distribute subtraction
x(x+2)=x²+2x, so the whole expression becomes x²+10x+24−x²−2x. Both terms of the second polynomial change sign.
Collect by power and inspect meaning
The x² terms cancel and the border area is 8x+24. A constant frame width often turns the area difference into a linear expression, which supports the model’s plausibility.

Expand a polynomial without changing the function it represents: An outer rectangle has dimensions x+6 and x+4, while an opening has dimensions x+2 and x. Its border area is (x+6)(x+4)−x(x+2). Simplify without choosing x.

Like terms are classified by their complete variable part

Terms 7x² and −3x² are like because both carry exactly x²; their coefficients combine to 4x². A term in x cannot merge with x², and a constant cannot merge with either. Standard form lists nonzero terms from the highest power down, making degree and leading coefficient immediately visible. Reordering uses commutativity, while regrouping uses associativity.

Parentheses require the distributive property. A minus sign before parentheses is multiplication by −1, so −(x−5)=−x+5. Exponents bind to their stated base: −x² means −(x²), while (−x)² means x². The direct algebra calculator can recognize several problem types; this focused page stays on polynomial expansion and combination.

A frame-width model rewards careful double distribution

An outer rectangle has dimensions x+6 and x+4, while an opening has dimensions x+2 and x. Its border area is (x+6)(x+4)−x(x+2). Simplify without choosing x.

Expand the outer product
(x+6)(x+4)=x²+4x+6x+24=x²+10x+24. Every term in one factor multiplies every term in the other.
Expand the opening and distribute subtraction
x(x+2)=x²+2x, so the whole expression becomes x²+10x+24−x²−2x. Both terms of the second polynomial change sign.
Collect by power and inspect meaning
The x² terms cancel and the border area is 8x+24. A constant frame width often turns the area difference into a linear expression, which supports the model’s plausibility.

Simplification, factorization, and evaluation preserve different information

Expanded standard form

It reveals degree and combined coefficients and is convenient for addition or coefficient comparison. It does not by itself identify zeros.

Factored form

It exposes multiplicative structure and can make roots visible through the zero-product property. The quadratic formula calculator remains reliable when a quadratic does not factor neatly over integers.

Numerical evaluation

Substituting a particular x returns one number. The exponent calculator can check a power within that arithmetic instance, but agreement at one value is evidence, not a proof that two symbolic forms are identical.

Polynomial-parser boundaries prevent invented algebra

Use explicit parentheses to preserve grouping, especially after a subtraction sign or around a powered sum.
Do not combine unlike powers, cancel across addition, or distribute an exponent over a sum: (x+1)² is not x²+1.
Radicals, named functions, inequalities, multiple variables, and unrestricted rational expressions are outside this focused grammar.
Compare the leading term and constant term before and after expansion; an unexpected degree increase or lost constant usually identifies the exact distribution step that failed.
If an equals sign asks for values of x rather than an equivalent expression, switch to the solve equation calculator for its declared linear form.

Important limitations

Simplify Calculator is limited to the finite inputs, modes, and outputs stated for the Interactive bounded polynomial simplifier method. Check polynomial expression, validation messages, the equation, and the stated display precision before relying on simplified polynomial, degree, and nonzero terms.

Review the stated scope, calculation method, assumptions, supported inputs, and effective dates, and obtain qualified advice before using a high-stakes result.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Polynomial rewrite bench

Expand a polynomial without changing the function it represents

Step-by-step guide · Polynomial rewrite bench

A frame-width model rewards careful double distribution

Variable scope: one variable, x
Canonical order: descending powers
Equality goal: equivalent expression, not a solved x

Expand the outer product
(x+6)(x+4)=x²+4x+6x+24=x²+10x+24. Every term in one factor multiplies every term in the other.
Expand the opening and distribute subtraction
x(x+2)=x²+2x, so the whole expression becomes x²+10x+24−x²−2x. Both terms of the second polynomial change sign.
Collect by power and inspect meaning
The x² terms cancel and the border area is 8x+24. A constant frame width often turns the area difference into a linear expression, which supports the model’s plausibility.

Like terms are classified by their complete variable part

A frame-width model rewards careful double distribution

An outer rectangle has dimensions x+6 and x+4, while an opening has dimensions x+2 and x. Its border area is (x+6)(x+4)−x(x+2). Simplify without choosing x.

Expand the outer product
(x+6)(x+4)=x²+4x+6x+24=x²+10x+24. Every term in one factor multiplies every term in the other.
Expand the opening and distribute subtraction
x(x+2)=x²+2x, so the whole expression becomes x²+10x+24−x²−2x. Both terms of the second polynomial change sign.
Collect by power and inspect meaning
The x² terms cancel and the border area is 8x+24. A constant frame width often turns the area difference into a linear expression, which supports the model’s plausibility.

Simplification, factorization, and evaluation preserve different information

Expanded standard form

It reveals degree and combined coefficients and is convenient for addition or coefficient comparison. It does not by itself identify zeros.

Factored form

Numerical evaluation

Polynomial-parser boundaries prevent invented algebra

Use explicit parentheses to preserve grouping, especially after a subtraction sign or around a powered sum.
Do not combine unlike powers, cancel across addition, or distribute an exponent over a sum: (x+1)² is not x²+1.
Radicals, named functions, inequalities, multiple variables, and unrestricted rational expressions are outside this focused grammar.
Compare the leading term and constant term before and after expansion; an unexpected degree increase or lost constant usually identifies the exact distribution step that failed.
If an equals sign asks for values of x rather than an equivalent expression, switch to the solve equation calculator for its declared linear form.

Expand the outer product

Expand the opening and distribute subtraction

Collect by power and inspect meaning

Like terms are classified by their complete variable part

A frame-width model rewards careful double distribution

Simplification, factorization, and evaluation preserve different information

Expanded standard form

Factored form

Numerical evaluation

Polynomial-parser boundaries prevent invented algebra

Important limitations

Expand the outer product

Expand the opening and distribute subtraction

Collect by power and inspect meaning

Like terms are classified by their complete variable part

A frame-width model rewards careful double distribution

Simplification, factorization, and evaluation preserve different information

Expanded standard form

Factored form

Numerical evaluation

Polynomial-parser boundaries prevent invented algebra